Optimal Path Partitions in Subcubic and Almost-subcubic Graphs

TL;DR

It is proved that the problem of partitioning the edges of a graph into as few paths into as few paths is fixed-parameter tractable when parameterized by the edge-deletion distance to a subcubic graph.

Abstract

We consider the problem of partitioning the edges of a graph into as few paths as possible. This is a~subject of the classic conjecture of Gallai and a recurring topic in combinatorics. Regarding the complexity of partitioning a graph optimally, Peroché [Discret. Appl. Math., 1984] proved that it is NP-hard already on graphs of maximum degree four, even when we only ask if two paths suffice. We show that the problem is solvable in polynomial time on subcubic graphs and then we present an efficient algorithm for ``almost-subcubic'' graphs. Precisely, we prove that the problem is fixed-parameter tractable when parameterized by the edge-deletion distance to a subcubic graph. To this end, we reduce the task to model checking in first-order logic extended by disjoint-paths predicates ($\mathsf{FO}\text{+}\mathsf{DP}$) and then we employ the recent tractability result by Schirrmacher, Siebertz, Stamoulis, Thilikos, and Vigny [LICS 2024].

Figures (3)

• Figure 1: An example of a pan cycle $abcde$ ($\deg(a)=3$ and $\deg(b)=\deg(c)=\deg(d)=\deg(e)=2$) and a bull cycle $uyvx$ ($\deg(u)=\deg(v)=3$ and $\deg(x)=\deg(y)=2$).
• Figure 10: Extending the path $P$ with the edge $vw$ reduces the number of used paths by one.
• Figure 11: The vertices in $V_4$ ($c$ and $h$) are gray squares, the ones in $N(V_4)$ are yellow disks labeled with the remaining letters, and the vertices in $V_X$ are pink disks labeled with numbers. The highlighted vertices form a terminal collection for the covering family comprising the five paths represented by the edge colors. The pattern contains the following traces: red $(x_2,b,c,d,x_5,x_4)$, blue $(x_1,a,c,e,x_3)$, purple $(x_5,f,h,k)$, brown $(g,h,i,x_6)$, orange $(h,j,x_7)$.

Theorems & Definitions (26)

• Theorem 1
• Lemma 1: Subcubic Path Partition $\bigstar$
• Corollary 3: GallaiGallai-2n-over-3-first
• Lemma 4
• Lemma 4: Subcubic Path Partition $\bigstar$
• Definition 5
• Lemma 5
• Lemma 6
• Definition 7: Bull-free Paths
• Lemma 8